Clear["Global`*"];

f[x_]:=Exp[x];

numTerms=5;

ts=Series[f[x],{x,0,numTerms}]

tsn=Normal[ts]

plot1=Plot[f[x],{x,-1,1},PlotStyle->Red]

plot2=Plot[tsn,{x,-1,1},PlotStyle->Blue]

plot3=Plot[f[x]-tsn,{x,-1,1},PlotStyle->Darker[Green],PlotRange->All]

MaxValue[{((x^6)/720),-1≤ x≤ 1},x]//N         (*  Estimste of the truncation error  *)

(*  Now with just 3 terms  *)

Clear["Global`*"];
f[x_]:=Exp[x];
numTerms=3;
ts=Series[f[x],{x,0,numTerms}]

tsn=Normal[ts]

plot1=Plot[f[x],{x,-1,1},PlotStyle->Red]

plot2=Plot[tsn,{x,-1,1},PlotStyle->Blue]

plot3=Plot[f[x]-tsn,{x,-1,1},PlotStyle->Darker[Green],PlotRange->All]

MaxValue[{((x^4)/24),-1≤ x≤ 1},x]//N                     (*  Estimste of the truncation error  *)